Resarchs on Solutions of Nonlinear Elliptic Equations and Numcrical Analysis

非线性椭圆方程解及数值分析研究

基本信息

批准号：
09440087
负责人：
YOTSUTANI Shoji
金额：
$ 3.14万
依托单位：
Ryukoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

项目摘要

Head investigator S.Yotsutani proposed a very accurate numerical computation method to solve Poisson's equation by using a charge simulation method with H.Morishita, N.Kobayashi, H.Takaichi and K.Amano. Recently, He has succeeded to shorten the time of the calculation considerably and improve the accuracy with H.Morishita and K.Anjano, This method is in the conformity with the paerallel computing. Thus it is possible to know the detailed shape of solutions of nonlinear elliptic systems.On the other hand, he developed the mathematical method of investigate the shape of radially symmetric solutions with Y.Kabeya and E.Yanagida. Recently, he has found the systematic change of variables to transform the differential equations arising from the elliptic equations to canonical form with E.Yanagida. Thus, relations between various equations studied one by one independently become very clear, and the understandings encourage the deep understanding of the propertites of each solution.The reserach results of investigators are asfollows. T.lkeda invesitigate the instabiliz ation of the standing pulse solutions of bistable reaction-diffusion systems. Y.Morita showed stabilization of vorticies in the Ginzburg-Landau equation with a variable diffusion coefficient. H.Oka investigated the connecting orbit structure of monotone solutions in the shadow system. Y.Yamada proved the coexistence states for some population models with nonlinear cross-diffusion. E.Yanagida discoverd various new phenomena of the systems of parabolic equations and gave the proof of them.

首席研究员 S.Yotsutani 与 H.Morishita、N.Kobayashi、H.Takaichi 和 K.Amano 一起提出了一种非常精确的数值计算方法，通过使用电荷模拟方法来求解泊松方程。最近，他与H.Morishita和K.Anjano成功地大大缩短了计算时间并提高了精度，该方法符合并行计算的要求。从而可以知道非线性椭圆系统解的详细形状。另一方面，他与Y.Kabeya和E.Yanagida一起发展了研究径向对称解的形状的数学方法。最近，他与E.Yanagida一起发现了变量的系统变化，将椭圆方程产生的微分方程转化为规范形式。这样，一个一个独立研究的各个方程之间的关系就变得非常清晰，这种理解促进了对每个解的性质的深入理解。研究者的研究结果如下。 T.lkeda 研究了双稳态反应扩散系统的常驻脉冲解的不稳定性。 Y.Morita 证明了具有可变扩散系数的 Ginzburg-Landau 方程中的涡流稳定性。 H.Oka 研究了阴影系统中单调解的连接轨道结构。 Y.Yamada 证明了一些具有非线性交叉扩散的群体模型的共存状态。 E.Yanagida发现了抛物方程组的各种新现象并给出了证明。